Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,18,12,21)(11,16,19,24,20,23,22,15);(5,6)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,21)(17,23)(18,24);(2,4)(6,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23);(6,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23);(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,24,16)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305332, #72305334, #72305338, … |
| maximal normal subgroup of | #72305332, #72305334, #72305338, … |
| maximal quotient of | #72305640, #72305757, #72305917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344978);